In Langlands' program, Satake correspondence gives a correspondence between unramified representation of a reductive group $G$ over a local field and conjugacy classes in the Langlands dual group ${}^{L}G$ whose projection to $\hat{G}$ is semisimple and projection to $W_K$ is Frob.
In page 11 of this article, there is similar but different correspondence. It gives a bijection between irreducible representations of $\mathrm{GL}(2, \mathbb{F}_{q})$ and conjugacy classes of $\mathrm{GL}(2, \mathbb{F}_{q})$. Also, the type of the conjugacy class (Jordan form) determines the type of representation (principal series, special, cuspidal, 1-dimensional).
Is there a general theory for such correspondence over finite field? Can we generalize this to arbitrary reductive groups over finite field? If it is, what is the correspondence? In the article, author said that the correspondence is kind of ad hoc, which is not canonical at all. However, if we fix a generator of $\mathbb{F}_{q}^{\times}$, than I think it may possible to find some canonical way to do it.
This is known as Deligne-Lusztig theory. See Delinge, Lusztig Representations of Reductive Groups Over Finite Fields.